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Capacitance in series and parallel circuits

What is capacitance?

“Capacitance is the amount of charge that a capacitor can store per unit of voltage across its plates.”It is denoted by C.That is capacitance is a measure of a capacitor’s ability to store charge.The more charge per unit of voltage that a capacitor can store,the greater its capacitance,its formula is given as:

C=Q/V

Where C is capacitance,Q is voltage,and V is voltage.We can also find charge Q and voltage V by rearranging the above formula as:

Q=CV

V=Q/C

 Unit of electrical capacitance:

Farad is the unit of capacitance.One Farad is the amount of capacitance when one coulomb of charge is stored with one volt across its plates.

Most capacitors that are used in electronics work have capacitance values that are specified in micro farad (µF) and pico farads(pF) .A micro farad is one millionth of a farad, and a pico farad is one trillionth of a farad.

Factors controlling Capacitance:

The capacitance of a capacitor depends on the following factors:

Plate Area:

plate area

Capacitance is directly proportional to the physical size of the plates as determined by the plate area,A.A larger plate area produces a larger capacitance ,and a smaller capacitance .Fig(a) shows that the plate area of a parallel plate capacitor is the area of one of the plates.If the plates are moved in relation to each other,as shown in fig(b),the overlapping area determines the effective plate area.This variation in effective plate area is the basic for a certain type of variable capacitor.

Plate separation:

plate seperation

`Capacitance is inversely proportional to the distance between the plates.The plate separation is designated d,as shown in fig(a). A greater separation of the plates produces a smaller capacitance ,as illustrated in fig(b).As previously discussed,the breakdown voltage is directly proportional to the plate separation.The further the plates are separated ,the greater the breakdown voltage.

Dielectric Constant:

As you know,the insulating material between the plates of a capacitor is called the dielectric.Dielectric materials tend to reduce the voltage between plates for a given charge and thus increase the capacitance.If the voltage is fixed ,more charge can be stored due to the presence of a dielectric than can be stored without a dielectric.The measure of a material’s ability to established an electric field is called dielectric constant or relative permittivity,symbolized by ∈r.

Capacitance is directly proportional to the dielectric constant.The dielectric constant of a vacuum is defined as 1 and that of air is very close to 1.These values are used  as a reference,and all other materials have values of ∈r specified with respect to that of a vacuum or air.For example,a material with ∈r=8 can result in a capacitance eight times greater than that of air with all other factors being equal.

The dielectric constant ∈r is dimensionless because it is a relative measure.It is a ratio of the absolute permittivity of a material ,∈r,to the absolute permittivity of a vacuum ,∈,as expressed by the following formula:

r=∈/∈

Formula of capacitance in terms of physical parameters:

You have seen how capacitance is directly related to plate area,A,and the dielectric constant,∈r,and inversely related to plate separation ,d.An exact formula for calculating the capacitance in terms of these three quantities is:

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C=A ∈r∈/d

where ∈= ∈r=∈r(8.85×10-12F/m)

Formula for total capacitance:

  • Series combination

when capacitors are connected in series,the total capacitance is less than the smallest capacitance value because the effective plate separation increases.The calculation of total series capacitance is analogous to the calculation of total resistance of parallel resistors.capacitor in series

This type of combination has the following characteristics:

  1. Each capacitor has the same charge across it.If the battery supplies  +Q charge to the left plate of the capacitor C1 due to induction -Q charge is induced on its right plate and +Q charge on the left plate of the capacitor C2 i.e

Q= Q1 +Q2 +Q3

2:The potential difference across each capacitor is different due to different values of capacitances.

3:The voltage of the battery has been divided among the various capacitors.Hence

V=V1 +V2 +V3

    = Q/C1 +Q/C2 +Q/C3

     =Q[ 1/C1 + 1/C2 + 1/C3]

V/Q=[ 1/C1 + 1/C2 +1 /C3]

4:Equivalent capacitance:

we can replace series combination of capacitors with one equivalent capacitor having capacitance Ceq i.e,

1/Ceq = 1/C1 + 1/C2 + 1/C3

  • Parallel combination:

When capacitors are connected in parallel,the total capacitance is the sum of individual capacitances because the effective plate area increases.The calculation of total parallel capacitance is analogous to the calculation of total series resistance.capacitance in parallel

In the above figure the left plate of each capacitor is connected to the positive terminal of the battery by a conducting wire.In the same way,the right plate of each capacitor is connected to the negative terminal of the battery.This type of combination has the following characteristics.

  1. Each capacitor connected to a battery of voltage V  has the same potential difference as:

V= V1 +V2 + V3

       2.The developed across the plates of each capacitor will be different due to different value of capacitances.

3.The total charge Q  supplied by the battery is divided among the various capacitors.Hence:

Q = Q1 +Q2 +Q3

Or  Q=C1V + C2V+C3V

       Q=V(C1 +C2+C3)

       Q/V=C1 +C2+C3

4.We can replace the parallel combination of capacitors with one equivalent capacitor having capacitance Ceq,such that

                                                                                                                        Ceq=C1 + C2 +C3

In case of n capacitors connected in parallel,the equivalent capacitance is given by:

                                                                                                                     Ceq=C1 +C2 +C3+……+Cn

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5.The equivalent capacitance of parallel combination of capacitors is greater than any of the individual capacitances.

Capacitance of parallel plate capacitor:

Consider a parallel plate capacitor.The size of the plate is large and the distance between the plates is very small,so the electric field between the plates is uniform.

capacitance of parallel plate capacitor

The electric field ‘E’ between the parallel plate capacitor is:

relation of parallel plate capacitor

Capacitance of cylindrical capacitor:

Consider a cylindrical capacitor of length L,formed by two coaxial cylinders of radii ‘a’ and ‘b’.Suppose L >>b ,such that there is no fringing field at the ends of cylinders.

capacitance of cylindrical capacitor

Let ‘q’ is the charge in the capacitor and ‘V’ is the potential difference between plates.The inner cylinder is positively charged while the outer cylinder is negatively charged.We want to find out the expression of capacitance for the cylindrical capacitor.For this we consider a cylindrical Gaussian surface of radius ‘r’ such that a<<b.

If ‘E’ is the electric field intensity on any point of the cylindrical Gaussian surface,then by Gauss’s law:

If ‘V’ is the potential difference between plates,then

relation of capacitance of cylindrical capacitor

This is the relation for the capacitance of a cylindrical capacitor.

Capacitance of a spherical capacitor:

relation of capacitance of spherical capacitor

Capacitance of an isolated sphere:

capacitance of isolated sphere

Energy stored in capacitor:

Consider a capacitor with the capacitance ‘C’ ,which is connected to the battery of emf ‘V’ .If ‘dq’ charge is transferred from one plate to other,then the work done ‘dW’ will be:

dW =V dq

This work done is stored in the form of electric potential energy ‘dU’

dU =V dq

When the capacitor is fully charged then the total energy stored is:

RELATION OF ENERGY STORED IN CAPACITOR

The energy stored in the capacitor is the energy store in the electric field between its plates.So,the energy stored can be expressed in terms of electric field strength ‘E’.

Expression of energy stored in the electric field between the plates of capacitor:

relation of energy

Energy density:

The energy density ‘μ’ is described as the energy stored ‘U’ per unit volume ‘V’ .mathematically it is expressed as:

RELATION OF ENERGY DENSITY

If any dielectric medium having dielectric constant ‘Ke‘ is placed between the plates of capacitor,then the expression of energy stored in the electric field of capacitor ‘U’ and energy density ‘μ’ will become:

energy density 2

Capacitance with dielectrics:

Consider a parallel plate capacitor which is connected with a battery of emf ‘V’ .Let ‘A’ is the area of each plate and ‘d’  is separation between the plates.

If ‘q’ charge is stored in the capacitor when there is vacuum or air as medium between the plates ,then:

q= CV

Where ‘C’ is the capacitance of the capacitor,which,for the case of parallel plate capacitor is expressed as:

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capacitance of dielectric

Michael Faraday,in 1937,investigated that if the space between the plates of parallel plate capacitor is filled with some dielectric medium ,then the charge stored in the capacitor increased to ‘q’.And hence ,the capacitance of the capacitance also increases to ‘C’.

Dielectrics:

Dielectrics are the insulating materials through which the electric current cannot pass easily,because these materials have very high value of electrical resistance.For example,paper,pyrex,polystyrene,transformer oil,pure water,silicon etc.

There are two types of dielectrics:

  1. Polar Dielectrics
  2. Non polar dielectrics

Polar dielectrics:

The dielectric materials which have permanent electric dipoles moment ‘p’ are called polar dielectrics.These materials consists of molecules which are permanent dipoles.

In the absence of external electric field,the polar molecules are randomly oriented.As the result,these materials have no net dipole moment.

When the external electric field is applied,then all dipoles tend to align themselves with external electric field.But the thermal agitation tends to keep the dipoles randomly oriented.Hence the partial alignment of electric dipoles is produced in a polar dielectric medium for specific field strength.However,the alignment of dipoles can be increased by increasing the external electric field and decreasing the temperature.

Non polar dielectrics:

non polar dielectric

The electric materials which don’t have permanent dipole moments are called non polar dielectrics.In the absence of external electric field,the atoms are neutral.When the electric field of strength ‘E‘ is applied ,then it tends to separate to positive and negative charges on the atoms of molecules.As the result,the atoms and molecules of dielectric become dipoles,called induced dipoles and this process is called electric polarization.

If the non polar dielectric materials is placed in a electric field having strength ‘E‘ then another electric field ‘E’ is produced due to polarization of medium.The electric field produced due to the polarization of dielectric is always opposite to the direction of external electric field.So,the net electric field in the region is ‘E=E-E’. Hence,the net electric field is reduced due to polarization of dielectric medium.

Effect of Dielectric medium on capacitance of capacitor:

When there is no dielectric medium between the plates of capacitor ,the stored charge ‘q’ in the capacitor can be find out using expression:

                                                       q=CV

But when the dielectric medium is placed between the plates of capacitor,then the net electric field and net potential difference between the plates is reduced due to electric polarization of dielectric medium.Therefore,the battery does work to transfer more charge to increase the potential difference between the plates.Therefore,the more charge is stored in capacitor,when a dielectric medium is placed between its plates and hence the capacitance of capacitor increased.

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